# Kármán-Howarth equation #fluid-mechanics #turbulence #derivation We start with the [[Navier-Stokes equations]]: ![[Navier-Stokes equations#^ec02dc]] - Multiplying both sides by $u_k'$ to get $ u_k'\frac{\partial u_i}{\partial t} + u_k'u_j\frac{\partial u_i}{\partial u_j} = -\frac{u_k'}{\rho}\frac{\partial p}{\partial x_i} + \nu u_k'\frac{\partial^2 u_i}{\partial x_j\partial x_j} $ ^c0a0e4 - Second term on the LHS can be expressed as $\begin{align}&\frac{\partial}{\partial x_j}(u_iu_ju_k') = u_iu_j\cancel{\frac{\partial u_k'}{\partial u_j}} + u_i\cancel{\frac{\partial u_j}{\partial x_j}}^{\textrm{incompressibility}}u_k' + \frac{\partial u_i}{\partial x_j}u_ju_k'\\\implies &\frac{\partial}{\partial x_j}\left(u_iu_ju_k'\right) = u_k'u_j\frac{\partial u_i}{\partial x_j}\end{align}$ - $\xi_j = x_j' - x_j\implies \frac{\partial}{\partial x_j} = \frac{\partial \xi_j}{\partial x_j}\frac{\partial}{\partial \xi_j} = -\frac{\partial}{\partial \xi_j}$ $\frac{\partial}{\partial x_j}\left(\overline{u_iu_ju_k'}\right)= -\frac{\partial}{\partial \xi_j}\left(\overline{u_iu_ju_k'}\right)$ - Correlation of the pressure and velocity $ -\frac{u_k'}{\rho}\frac{\partial p}{\partial x_i} = \frac{1}{\rho}\frac{\partial}{\partial \xi_i}(p u_k') $ [[Pressure-velocity correlations]] $\overline{pu_k'} = 0\implies \frac{1}{\rho}\frac{\partial}{\partial \xi_i}\left(\overline{pu_k'}\right) = 0$ - $\frac{\partial^2 (u_iu_k')}{\partial x_j\partial x_j} = \frac{\partial^2 (u_iu_k')}{\partial \xi_j\partial \xi_j}$ - $\overline{u_k'\frac{\partial u_i}{\partial t}} - \frac{\partial}{\partial \xi_j}\left(\overline{u_iu_ju_k'}\right) = \nu\frac{\partial^2(\overline{u_iu_k'})}{\partial \xi_j\xi_j}$ - In an analogous way, we obtain $\overline{u_i\frac{\partial u_k'}{\partial t}} + \frac{\partial}{\partial \xi_j}\left(\overline{u_iu_j'u_k'}\right) = \nu\frac{\partial^2(\overline{u_iu_k'})}{\partial \xi_j\xi_j}$ - $\overline{u_iu_j'u_k'} = -\overline{u_i'u_ju_k}$ - $\overline{u_i\frac{\partial u_k'}{\partial t}} - \frac{\partial}{\partial \xi_j}\left(\overline{u_i'u_ju_k}\right) = \nu\frac{\partial^2(\overline{u_iu_k'})}{\partial \xi_j\xi_j}$ - $\frac{\partial}{\partial t}\left(\overline{u^2}R_{ik}\right) - \left(\overline{u^2}\right)^{3/2}\frac{\partial}{\partial \xi_j}\left(T_{ijk} + T_{kji}\right) = 2\nu\overline{u^2}\frac{\partial^2 R_{ik}}{\partial xi_j\partial \xi_j}$ - Rewrite tensors $R, T$ in terms of their defining scalars $f, g, k, q, h$. - $\frac{\partial}{\partial t}(\overline{u^2}f) + 2\left(\overline{u^2}\right)^{3/2}\left(\frac{\partial h}{\partial r} + \frac{4}{r}\right) = 2\nu\overline{u^2}\left(\frac{\partial^2 f}{\partial r^2} + \frac{4}{r}\frac{\partial f}{\partial r}\right)$